The Annals of Statistics

Tailor-made tests for goodness of fit to semiparametric hypotheses

Peter J. Bickel, Ya’acov Ritov, and Thomas M. Stoker

Full-text: Open access


We introduce a new framework for constructing tests of general semiparametric hypotheses which have nontrivial power on the n−1/2 scale in every direction, and can be tailored to put substantial power on alternatives of importance. The approach is based on combining test statistics based on stochastic processes of score statistics with bootstrap critical values.

Article information

Ann. Statist., Volume 34, Number 2 (2006), 721-741.

First available in Project Euclid: 27 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 62G09: Resampling methods

Copula models mixture of Gaussians independence


Bickel, Peter J.; Ritov, Ya’acov; Stoker, Thomas M. Tailor-made tests for goodness of fit to semiparametric hypotheses. Ann. Statist. 34 (2006), no. 2, 721--741. doi:10.1214/009053606000000137.

Export citation


  • Bickel, P. J. and Chernoff, H. (1993). Asymptotic distribution of the likelihood ratio statistic in a prototypical non-regular problem. In Statistics and Probability: A Raghu Raj Bahadur Festschrift (J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. L. S. Prakasa Rao, eds.) 83--96. Wiley, New York.
  • Bickel, P. J., Götze, F. and van Zwet, W. R. (1997). Resampling fewer than $n$ observations: Gains, losses and remedies for losses. Statist. Sinica 7 1--31.
  • Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD.
  • Bickel, P. J. and Ren, J.-J. (1996). The $m$ out of $n$ bootstrap and goodness of fit tests with doubly censored data. Robust Statistics, Data Analysis, and Computer Intensive Methods. Lecture Notes in Statist. 109 35--47. Springer, New York.
  • Bickel, P. J. and Ren, J.-J. (2001). The bootstrap in hypothesis testing. In State of the Art in Probability and Statistics. Festschrift for Willem R. van Zwet (M. de Gunst, C. A. J. Klaassen and A. van der Vaart, eds.) 91--112. IMS, Beachwood, OH.
  • Bickel, P. J. and Ritov, Y. (1992). Testing for goodness of fit: A new approach. In Nonparametric Statistics and Related Topics (A. K. Md. E. Saleh, ed.) 51--57. North-Holland, Amsterdam.
  • Bickel, P. J., Ritov, Y. and Stoker, T. (2005). Nonparametric testing of an index model. In Identification and Inference for Econometric Models: A Festschrift in Honor of Thomas Rothenberg (D. W. K. Andrews and J. H. Stock, eds.). Cambridge Univ. Press.
  • Choi, S., Hall, W. J. and Schick, A. (1996). Asymptotically uniformly most powerful tests in parametric and semiparametric models. Ann. Statist. 24 841--861.
  • Durbin, J. (1973). Distribution Theory for Tests Based on the Sample Distribution Function. SIAM, Philadelphia.
  • Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153--193.
  • Feuerverger, A. and Mureika, R. A. (1977). The empirical characteristic function and its applications. Ann. Statist. 5 88--97.
  • Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. Academic Press, New York.
  • Hart, J. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York.
  • Ibragimov, I. A. and Hasminskii, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, New York.
  • Ingster, Yu. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III. Math. Methods Statist. 2 85--114, 171--189, 249--268.,
  • Janssen, A. (2000). Global power functions of goodness-of-fit tests. Ann. Statist. 28 239--254.
  • Kac, M., Kiefer, J. and Wolfowitz, J. (1955). On tests of normality and other tests of goodness-of-fit based on distance methods. Ann. Math. Statist. 26 189--211.
  • Kallenberg, W. and Ledwina, T. (1999). Data-driven rank tests for independence. J. Amer. Statist. Assoc. 94 285--301.
  • Khmaladze, E. V. (1979). The use of $\omega^2$ tests for testing parametric hypotheses. Theory Probab. Appl. 24 283--301.
  • Klaassen, C. A. J. and Wellner, J. A. (1997). Efficient estimation in the bivariate normal copula model: Normal margins are least favourable. Bernoulli 3 55--77.
  • Neyman, J. (1959). Optimal asymptotic tests of composite statistical hypotheses. In Probability and Statistics: The Harald Cramér Volume (U. Grenander, ed.) 213--234. Almqvist and Wiksell, Stockholm.
  • Pfanzagl, J. (with the assistance of W. Wefelmayer) (1982). Contributions to a General Asymptotical Theory. Lecture Notes in Statist. 13. Springer, New York.
  • Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York.
  • Rayner, J. C. W. and Best, D. J. (1989). Smooth Tests of Goodness-of-Fit. Oxford Univ. Press, New York.
  • Roy, S. N. (1957). Some Aspects of Multivariate Analysis. Wiley, New York.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Wald, A. (1941). Asymptotically most powerful tests of statistical hypotheses. Ann. Math. Statist. 12 1--19.